Related papers: On $b$-Whittaker functions
We consider eigenfunctions of many-body system Hamiltonians associated with generalized (a-twisted) Cherednik operators used in construction of other Hamiltonians: those arising from commutative subalgebras of the Ding-Iohara-Miki (DIM)…
We build in this paper the algebra of q-deformed pseudo-differential operators shown to be an essential step towards setting a q-deformed integrability program. In fact, using the results of this q-deformed algebra, we derive the…
In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the…
We repeat, using methods developed for BKM systems, the famous results of S. Novikov (1974), J. Moser (1981, 1982) , and A. Veselov (1980) that relate Schr\"odinger-Hill operators with finite-band spectra, solutions of the Neumann system,…
We prove that cuspidal automorphic D-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from GL_n to general reductive groups. The key tool is a microlocal interpretation of…
We introduce and study invariant differential operators acting on the space $\mathcal{H}(\Omega)$ of holomorphic functions on the complement ${\Omega=\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w \not=1\}}$ of the "complexified unit…
We provide a formulation of quantum mechanics based on the cohomology of the Batalin-Vilkovisky (BV) algebra. Focusing on quantum-mechanical systems without gauge symmetry we introduce a homotopy retract from the chain complex of the…
Studied is the Baxter equation for the quantum discrete Boussinesq equation. We explicitly construct the Baxter $\mathcal{Q}$ operator from a generating function of the local integrals of motion of the affine Toda lattice field theory, and…
Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld's upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note we show that the…
Considering the kernel of an integral operator intertwining two realizations of the group of motions of the pseudo-Euclidian space, we derive two formulas for series containing Whittaker's functions or Weber's parabolic cylinder functions.…
We investigate an N-state spin model called quantum relativistic Toda chain and based on the unitary finite dimensional representations of the Weyl algebra with q being N-th primitive root of unity. Parameters of the finite dimensional…
Using results from theory of operators on a Hilbert space, we prove approximation results for matrix-valued holomorphic functions on the unit disc and the unit bidisc. The essential tools are the theory of unitary dilation of a contraction…
We introduce a variational approach for the Quantum Inverse Scattering Method to exactly solve a class of Hamiltonians via Bethe ansatz methods. We undertake this in a manner which does not rely on any prior knowledge of integrability…
We conjecture an explicit construction of integral operators intertwining various quantum Toda chains. Compositions of the intertwining operators provide recursive and Q-operators for quantum Toda chains. In particular we propose a…
Several new formulas are developed that enable the evaluation of a family of definite integrals containing the product of two Whittaker W-functions. The integration is performed with respect to the second index, and the first index is…
A general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh-Weyl $m$-function is proved. This result is applied to scattering…
We continue the study of the B-Toda hierarchy (the Toda lattice with the constraint of type B) which can be regarded as a discretization of the BKP hierarchy. We introduce the tau-function of the B-Toda hierarchy and obtain the bilinear…
We study Whittaker vectors (and Jacquet integrals) in the generalized principal series for a real reductive group. A functional equation for them is obtained. This allows to establish uniform estimates for their holomorphic extensions with…
In this paper, a spectral theorem is proved for self-adjoint cyclically compact partial integral operators in the space of functions with mixed norm, which is a Kaplansky--Hilbert module. The decomposition through eigenfunctions, integral…
We consider the Hamiltonian reduction of the two-loop Wess-Zumino-Novikov-Witten model (WZNW) based on an untwisted affine Kac-Moody algebra $\cgh$. The resulting reduced models, called {\em Generalized Non-Abelian Conformal Affine Toda…